"       26   '^C 


SOME  THEOREMS  ON  THE 
SUMMATION  OF  DIVERGENT  SERIES 


DISSERTATION 


SUBMITTED   IN    PARTIAL  FULFILLMENT   OF   THE   REQUIRE- 
MENTS FOR  THE  DEGREE  OF  DOCTOR  OF  PHILOSOPHY, 
IN   THE   FACULTY   OF   PURE   SCIENCE, 
COLUMBIA   UNIVERSITY. 


BY 

GLENN  JAMES 


PRINTED    BY 

W.  D.   Gray,  227  West  17th  Street 

New  York  City 

1917 


SOME  THEOREMS  ON  THE 
SUMMATION  OF  DIVERGENT  SERIES 


DISSERTATION 


SUBMITTED   IN    PARTIAL  FULFILLMENT   OF   THE   REQUIRE- 
MENTS FOR  THE  DEGREE  OF  DOCTOR  OF  PHILOSOPHY, 
IN    THE    FACULTY    OF    PURE    SCIENCE, 
COLUMBIA   UNIVERSITY. 


BY 

GLENN  JAMES 

w 


PRINTED   BY 

W.  D.   Gray,  227  West  17th  Street 

New  York  City 

1917 


-S3 


SOME  THEOREMS  ON  THE  SUMMATION  OF  DIVERGENT  SERIES 

by 
GLENN  JAMES. 


1.     The  Definition  of  the  Sum  of  a  Divergent  Series. 

The  various  definitions  of  the  sum  of  a  divergent  series  are  based  directly  or 

n 
indirectly  upon  the  use  of  so  called  convergence  factors.*     A  series  2  a.p  is  trans- 
formed into  1 

n 

(1)  2      apfp  (x„X2,  ...,Xfc) 

vfhevG  if,  (Xi,  X;,,  ....  xj.)  is  such  that  the  "subsidiary  sequence,"  (1),  converges 

00 

for  all  values  of  x^,  x^,  . . .,  x^  on  certain  specified  ranges.     The  sum  of      2      a, 
is  then  defined  to  be  f  P  =^  ^ 

n 

(2)  S=       L  L         ...        L  L  2    ap  fp  (xi,  Xg,  .. .,  Xfc) 

Xi  ->  Li  Xj  -^  Lj  Xfc  — »  Lt  n  ->  CO  p  =  1 

For  brevity  we  will  write  the  above  as  follows : 

n 

(3)  S=        L  L       2,  apfp  (x<),  (i=  1,2,  3,  ...,k) 

x<  — >  Li  n  — »  00    1 

For  example  the  series 
can  be  summed  by  taking, 

1 

fp  (xi)  = ,  X  >  0,  c>  1 

f.px 

Then 

n  (— l)p*^                         1 
S=        L  L     2 =       L     =  0 


X— >oon— >ool       c"*  X— >ooc*-|-l 

We  might,  however,  choose 


*  C.  N.  Moore:     Trans.  Am.  Math.  Soc.  (1901),  p.  300. 
S.  Chapman:     Quarterly  Journal  (1912),  V.  43,  p.  1. 

L.  L.  Smail:      Dissertation,  "Some  Generalizations  on  the  Theory  of  Summable 
Series,"  Columbia  University  (1913),  Chap.  II,  p.  4. 
t  We  may  have  Xj.=n  as  in  Cesaro's  mean,  §  3.      In  this  case  there  would  be  only  k 
limiting  processes. 


371JJ0 

3 


.  1 

fp(Xi)=  ,  X>0,  C>1 

qP/M 

In  this  case  S  =  3^. 

Evidently  the  usefulness  of  the  general  definition  depends  upon  the  restric- 
tions which  are  placed  upon  fp(xi). 

(a)  Consistency  in  the  general  theory  of  series  demands  that  any  method 
of  summing  divergent  series  should  be  such  as  to  give  the  ordinary  sum  when 
applied  to  convergent  series.  This  is  called  the  "condition  of  consistency."  *  (It 
is  evident  that  the  first  method  in  the  above  example  does  not  satisfy  this  condi- 

1 

tion,  since       L      =  0.     But  the  second  method  does  since, 

X  — ^  00    c"' 

1  11 

L      =  1 ;       L      =  0 and is  always  positive.)  f 

X  -^  00    C^'''^  p  — >  00   cP''  c^'* 

(b)  Evidently  the  sum  of  any  series  should  be  the  function  whose  expan- 
sion gives  the  series,  when  such  a  function  exists  uniquely.^ 

(c)  When  we  are  summing  a  series  of  variable  terms,  which  converges 
uniformly,  it  is  desirable  that  the  "subsidiary  sequence"  should  converge  uniformly 
with  respect  to  the  variable  of  the  original  series.  If  such  be  the  case,  a  series  is 
said  to  be  "uniformly  §  summable"  by  the  given  method. 

In  general  it  seems  desirable  to  seek  methods  of  summing  divergent  series 
which  make  the  sum  bear  the  closest  possible  analogy  to  the  ordinary  sum  of 
convergent  series.  However,  in  this  paper  we  shall  be  concerned  only  with  the 
above  restrictions,  which  are  indeed  most  essential. 

In  the  next  article  a  mean  for  summing  divergent  series  is  developed  which 
satisfies  the  condition  of  consistency  and  sums  uniformly  convergent  series  uni- 
formly. This  mean  includes  the  most  interesting  form  of  Euler's  Transforma- 
tion for  slowly  convergent  series  ^  as  a  special  case,  and  with  a  slight  modification 
becomes  a  generalization  of  the  means  of  Frohenins  and  Holder. 

Properly  divergent  series  have  been  shown  to  be  non-summable  by  the  well 
known  methods  of  Borel,  LeRoy,  Cesaro  or  Holder.  ||  The  third  article  of  this 
paper  extends  the  notion  of  proper  divergence  to  a  class  of  oscillating  series 
which  are  non-summable  by  the  above  methods  and,  in  fact,  by  the  above  general 
definition  of  p.  1,  when  certain  restrictions  are  placed  upon  it.     These  restrictions 


*  Bromwich:     "Theory  of  Infinite  Series,"  p.  269. 
t  See  article  3,  Theorem  VIII. 
tPringsheim:     Encyklopedie,  Bd.  I,  A.  3,  39. 

§  Any    series  is  said  to  be  "uniformly  summable"  when  the  subsidiary  sequence  con- 
verges uniformly;  see  Hardy,  Transactions  Cambridge  Philos.  Soc,  Vol.  19  (1904),  p.  301. 
^  Bromwich  :      Loc.  cit.,  p.  55. 
(l  L.  L.  Smail :    Dissertation,  Columbia  University  (1913),  Chap.  III. 


are  shown  to  make  the  definition  satisfy  the  condition  of  consistency  and  to  sum 
uniformly  convergent  series  uniformly. 

2.     A  Mean  for  Summing  Series. 

Consider  the  series 

(4)  ai  +  ag  +  aa  +  . . .  4-  ar  =  Sr 
where 

L      Sr  =  s 
r  — >  00 

The  plan  followed  in  the  first  part  of  the  succeeding  development  is  as  fol- 
lows: Take  the  means  m  at  a  time,  of  the  terms  of  (4)  beginning  with  the  first, 
then  the  second,  and  so  on;  make  corrections  to  maintain  the  identity;  repeat 
the  process  on  the  new  series  n-1  successive  times  and  take  the  limit  as  r  increases 
indefinitely. 

For  n  5  1  and  r  5  m,*  we  have, 

(5)  ai+a2+a3+...+a„       a^+^z+^i-h  • -- +^m^x      ^3+^4+^5+- -  •+^m*2 


m  m 


-f  ...+ 


l+ar_m+2+ar.n,+3+ar  (m— 1  )ar+ (m— 2)ar.i4- •  •  ■+^r.m^2 


m  m 

(m — 1  )a,4- (m — 2)a2+ . . .  +am-i 


—  Sr- 


m 


The  sum  of  the  first  m  terms  of  (5)  is, 

(ai  +  2a2  +  3a3  +  ...  -f-  m  a„,  -f-  ...  -f-  agm.J/m 

From  the  expansion,  by  actual  multiplication,  of 

(1  +x  +  x2-f  ...  -f  x»"-^)' 

we  see  that  the  above  numerator  is  just  this  expansion   with  x'  replaced  by 
a^+i,  q  =  0,  . . .,  (2m  —  2).     Hence  we  will  denote 

(ai  +  2a2  +  3a3  +  . . .  +  ma^  H f-  a-zm-^) 

by 

'(a,  +  ...  +a«) 

Taking  the  means  of  (5)  we  now  obtain,  with  this  notation: 

(6)     + +---  + 

m*  m^  m^ 


*  Although  this  and  the  following  identities  hold  for  m=:l,  this  is  a  trivial  case. 
Hence  it  will  be  understood  throughout  this  paper  that  m>l. 


(m — l)ar+(m— 2)ar_i+. .  .+ar.m+2 

+ + 


m 


(m— 1  )  (ar+  .  .  .  4-ar_m^i)  +  (m— 2)  (ar.i+  .  .  .  +ar.m)  +    (ar_m,2+  •  •  •  +ar-om,3  ) 


=  Sr 


(m— l)ai+(m— 2)a2+.  .  .+a,„.i 


m 


(m— 1)  (ai+. .  .+a„.)  +  (m— 2)  (a^-\-..  .+am.i)  +  -  •  .  +  (am-i4--  •  -+^2^-^^ 


Evidently  the  sum,  as  before,  of  the  first  m  terms  of  (6),  r  being  chosen  suf- 
ficiently large,  will  give  a  form  in  a^,  a^-  •  -,  ^3m.2  which  is  just  (1+x-j-. .  .^-x^'O^ 
with  x«  replaced  by  a^+i,  (q=0,  1,  2  ...  3m — 3).  Analogous  to  the  procedure 
above  we  denote  this  form  by  ^{a..^-}-. .  .+am).  Similarly  the  nth  mean  will  give 
us  forms  "(ai+. .  .+am),  which  are  equivalent  to  (l+x4-.  •  •+x'""^)"  with 
x^^rag+i,  q:=0,  1,  ...,  n(m — 1)  since  there  are  n(m — l)  +  i  terms  in  the  nth 
power  expansion  of  the  sum  of  m  terms. 

We  can  now  write  out  the  identity  resulting  from  the  n — 1th  repetition  of  the 
above  process.     It  is  as  follows : 


(7) 


"(ai+--.+am)       "(a2+...+am,i) 


m" 


+  ...+ 


y^r-n(m-i)~T'  • '  •  ~T~ar_n(m_i)+in_i  )     "r  -K  r 


m" 


=  Sr 


(m— l)ai-f(m— 2)a2+-  -.+3^-1 


+ 


m 


(m— 1 )  (a,  + . . .  H-a„,)  +  (m— 2)  (aa-f . . .  +a^, J  + . . .  +  (a,„_i+ . . .  +a„„., ) 


+  ...+ 


(m— l)«-^(a,+  ...+a„)  +  (m— 2)"~M;a,+  ...+a„„i)^ 


m' 


. . .  -]-"    {a.m-1-f- .  .  .  -f-agm-o  ) 


where, 

(m — l)ar+(m — 2)ar_i+. .  .+ar_m+2      (^ — l)(ar+. .  .+ar_m+i)4- 

RV=: + 

m  m^ 

(m 2)(ar_i+..  .+ar.m)  +  .  .  .+(ar_m+2+--  •H-ar-am+s) 


+  ...+ 


■(m— 1)  "-Har+-  •  .+ar.„,.J  +  (m— 2)  »-i(a..,+  . .  .+a,._„.)  +  .  • 


~r"'  (ar_m+2~r' •  •~rar_2f»+3) 


and  r^n  (m — 1)  +  1. 

We  will  now  seek  the  limiting  form  of  (7),  as  r  increases. 

Consider  first  the  remainder  R',-. 

Since  the  series    (4)    converges,   for  every  c,  there  exists  an   r'   such  that 
for  r  5  r' 

_  £5  la,._,!,   [iz=0,  1,  2,   ...,   (n— l)(m— l)+m— 2] 
whence  for  r  >  r' 


RV5 


(m— 1 )  +  (m— 2)  + ...  +1        (m— 1  )m+  (m— 2)m+ . . .  +m 


+ 


(m— l)m"-i-|-(m— 2)m»-i+. .  .+m"-^ 


m» 


n(m— 1) 


and  it  follows  that 

(8)         L     R'r=0,  for  every  n  and  m. 
r— >oo 


The  terms 


«(ai+. .  .H-a„,)   «(a2+. .  .+a„,,i)  «(ar_n(m-i)  +  .  •  .+ar.(„.i)(m-i)), 


m" 


m" 


when  added  term  by  term  take  the  form 

(9)  "[(ai+-  •  •+ar-«(m-i))  +  (a2+.  .  .  +ar_„(m_x).i)  +  .  .  • 

+  (am-f- •  •+ar_(n-i)(m-i))]l/m".     This  can  be  written, 

(10)  "[Sr_n(»i-i)+ (S;_n(m-i)+i       Si)-)--  •  •  +  (S,_(n_i)  (m-i)       Sm_i)]l/ni'* 

The  last  term  in  the  expansion  of  ( 10)  would  be 

Sr_n  ( m-i )  +n  (m_i)  +i       Sn(m-i)^^Sr+i       Sn  (m_i ) 
CO 

Now  since  2  a„  =  s 
1 


(11^  •L'     Sr-mm-i) —       •L'       Sr_(>H_i)+i — ... —       L.     Sj- —       L     Sr+j^ — S, 

r— »oo  r— ^00  r— ^00        r-^oo 

m  and  n  being  fixed. 
Whence 

(12)  L     "[Sr-«(m-x)  +  (Sr-»(m_i)+i— Si)  +  .  .  .4-(Sr.(«.i)(m-i)— Sw-i)]l/m» 

r— »oo 

="[s+(s— sj  +  . .  .  +  (8— s„,.,)]l/m» 

="(Rc,+Ri+...+R„..,)l/m» 

where  Ro=s  and  Ri  is  the  remainder  after  the  ith  term. 

From  (8),  (12)  and  the  fact  that      L    Sr=s,  we  see  that  the  limiting  form  of 

r— >oo 

(7)  is 

(m— l)ai+...4-a„,.j 

(13)  «(R<,+  ...4-Rm_01/ni"=s + 

L  1^1 

"-Hm—l)*(a,+  ...+a„..,)  +  (m— 2)  *(a2+..+a.„)  +  ..+*(a»<.,4-- ••+».»,-,) 

2 . 

i.i  m**i 

or 

(14)  "(R„+. .  .+R;„.,)l/m"r=.s— „,S'„ 

where  mS'»  denotes  the  second  term  of  the  right  member  of  (13). 

Theorem  I.     When  the  series  aj+a2+. .  .+a«+.  •  •  converges  to  s, 

n— >oo 

To  prove  the  theorem,  it  is  sufficient  to  show  that  the  left  member  of  (14) 
converges  to  zero. 

Expand  "(R„+. .  .+R,„_i)l/m'»,  denoting  the  coefficients  by  Co,  c^,  . .  .,  c„(,„-i) 
and  write  the  expansion  as  follows : 

Ro+CiRi-f- . . .  +c/,-Rfc    Cfc^iRfr^iH- . . .  +R„(m.i) 

(15)  + , 

m"  m** 

noting  that  Co=c„(„,.i)  =  l. 

Now  for  every  e  there  exists  a  (k)  such  that 

(16)  lR)i-.x|^c/2,  i=l,2,3,... 

00 

since  2a„:=s. 
1 

Whence 

(17)  |CA-,,R..,+  .-.+R«<m-„|l/m'' 

^«/2[q,,+  .  .  .+c.(m-a).i+]l/m"5e/2,  n(m— l)>k 
8 


Consider  the  first  k-f-1  terms  of  (15). 

Since       L    l/m"=    L     Ci/m"=...=     L    ct/m»=o 
n— >  00  n-^  00  n— >  oo 


there  exists  an  N  such  that 

(18)  lR<,+c,R,+  .?.4-CfcRfcl 


ge/2,  n>N. 


m" 
Combining  (17)  and  (18)  we  have 

(19)  l''(Ro+. .  .+Rm-i)l/ni"l  ^e,  n>N',  N'  being  equal  to  or  greater  than 

k 

either or  N.     Thus  the  theorem  is  proved. 

m— 1 

From  (12)  and  (14). 

(20)  s— ;„S'„="[s+(s— s,)  +  . .  .  +  (s— s..a)]l/m" 

="(s+s+...+s)l/m»— »(0+s,+  ...+s,„.i)l/m'» 
=s— "(0+s,+ . . . H-s^.i) l/m» 

where  ,„S„rr:(0+Si+. .  .+Sm_i)l/m". 

Then 

(21)  t«S„=:mS'„  or  written  out  in  full 

n(n±l)* 

(22)  (n  SiH s,+ . . .  +Sn(m.i) )  1/m" 

2! 

=  [(m— l)a,+  (m— 2)a,+  ...+a„_Jl/m+ 
n— 1  r  (m— 1)  *(ai+...+a^)  +  *(a2+...+a„,^i)  +  ...+<(a;„.i  +  ...+a2„.2) 


i=l 


m* 


Theorem  II.     When  the  series  3^4-^2+^3+-  •  •  converges  to  s 

n-^oo  m— »oo  m-^00  m,n— >oo  m,n— »oo 

That      L      ,„S„=s  follows  from  (21)  and  Theorem  I.* 
n— >oo 

Consider       L     ,„S„=     L     mS'n—s.     The  proof  of  Theorem  I  suffices   for 
m— »  00  m— >  00 

these  cases  provided  we  fix  n  and  let  m  vary,  replace  the  conditions  n>N  and 
n>N'  by  m>M  and  m>M'  and  note  in  connection  with  (18)  that  the  first  k 


The  plus  sign  occurs  for  m>.2. 


coefficients  in  the  expansion  of  (1+x-f . .  .+x'"-^)'»  never  change  with  m  provided 
m>K. 


Finally  consider      L      ,„S„: 
m,n— »oo 


L       mS'„=s.     Since  n  and  m  both  increase, 
m,n-^co 


replace,  in  Theorem  I,  the  conditions  n>N  ancf  m>]Vr  by  the  pairs  n>N,  m>M 
and  n>N',m>M',  respectively,  N'  and  M'  having  been  chosen  so  that,  (m — l)n  5  k 
and  the  inequality  (18)  holds  for  n>N'  and  m>M'.  This  proof  then  suffices 
for  the  above  cases. 

Theorem  III.  When  a  series  of  variable  terms  converges  uniformly  it  is 
uniformly  summable  by  mSn  and  mS',,  with  either  n  or  m  or  both  increasing. 

The  proofs  of  Theorems  I  and  II  suffice  for  this  theorem  when  (16)  is 
replaced  by  (Rt+i)  ^  e/2,  i=l,  2,  3,  ....  for  all  values  of  the  independent  variables 
on  the  intervals  considered.  This,  of  course,  is  just  the  definition  of  uniform  con- 
vergence. 

Thus  it  has  been  proven  that  the  condition  of  consistency  is  satisfied  and 
uniformly  convergent  series  are  uniformly  summable  by  ,„S„  and  „,S'„  with  m  fixed 
and  n  increasing  or  with  n  fixed  and  m  increasing  or  with  both  m  and  n  increas- 
ing in  any  manner  whatever,  m  and  n  being  on  the  range,  (1,  2,  3  . . . ). 

Upon  the  basis  of  „,Sn  we  define  the  two  following  sets  of  means : 

(A) 

n(n±l)s2 

n  Si-| f-  .  .  .  -}-S„(m_i) 


S'^' 


S'^' 


,S':? 


2! 
n(n±:l) 

2! 

n(n±:l) 


l/m"=„,S„ 
1/m" 


Hmo     1 


2! 


S['--i]4_  I Q    [r-D 


l/m" 


where  m  is  fixed  and  n  increases. 

(B) 

n(n±l) 


nsi-l- 


2! 


S2"T"  •  •  •  ~\~Si7um-i) 


}S,r 


':^s,.= 


•     n(nzt:l) 

n  1  «Jw  n  2 '-'"  n^  •  •  •  1  //  ()„.i)j  „ 


n    1  ^)i 


2! 
n(n±l) 


l/m"=rt,S„ 
l/m" 


(r-UC     4_  I        [;•  I]    C 


l/m** 


.    2! 

where  n  is  fixed  and  m  increases. 

Theorem  IV.     The  means  (A)  and  (B)  satisfy  the  condition  of  consistency 

10 


for  all  values  of  the  parameters  m  and  r  in  (A)  and  n  and  r  in  (B). 

Consider  first  the  means  (A).     We  have  proved  (Theorem  I)  that  the  above 
theorem  is  true  when  r=:l. 
Hence  when 

(23)     Sj,  Sj,  ...  Sn  ...  converges  to  s  so  does 

(24)   „,s'i\  „s^ ,„s':,' ... 

Applying  Theorem  I  to  ,nS^^\  we  see  that  the  sequence 

cr2]       C[2]  c[2) 

converges  to  s  whenever  (24)  does,  hence  whenever  (23)  does. 

The  theorem  follows  by  induction. 

The  argument  for  (B)  is  the  analogue  of  this  since  the  theorem  is  proven  foi 
r=:l  in  Theorem  II. 

Corollary.  If  (A)  or  (B)  sums  a  series  for  any  value  of  r.  it  sums  th< 
series  to  the  same  sum  for  every  greater  value  of  r. 

Theorem  V.  The  means  (A)  and  (B)  sum  uniformly  convergent  series 
uniformly. 

This  theorem  has  been  proven  (Theorem  III)  for  r=l.  The  proof  for  any  i 
is  the  same  as  the  proof  of  Theorem  IV,  except  that  the  convergence  of  eacl 
sequence  is  uniform. 

The  means  (A)  and  (B)  can  not  sum  properly  divergent  series. 

This  becomes  evident  for  rr=l  when  we  observe  that  for  every  k  the  sum  o 
the  last  n(m — 1)  +  1 — k  coefficients  of  (l+x+. .  .+x"'-^)"  divided  by  m",  the 
sum  of  all  of  them,  has  the  limit  unity.  The  arguments  for  „,S»f'''  and  ^''^mS,,  ar( 
the  same  as  that  of  Theorem  IV  except  that  we  have  proper  divergence  of  eacl 
sequence  instead  of  convergence. 

If  n  and  m  increase  together,  in  any  manner  whatever,  in  (A)  and  (B) 
Theorem  IV,  its  corollary,  Theorem  V  and  the  above  remark  on  proper  diver 
gence  hold  for  these  methods.  This  follows  from  Theorem  IT  and  Theorem  II 
by  arguments  similar  to  those  used  for  (A)  and  (B). 

Euler's  Transformation  for  Slowly  Convergent  Series,  with  x=l,*  is  a  specia 
instance  of  „,S'„  namely  gS',,,  hence  is  identically  equal  to  .S',)\  (=2S„).  Fo 
putting  m=2  in  (22)  gives 

(25) 

n(n — 1)  1      aj      a^+a-      a-,-f-2a.,+a., 

,S„=(n  s,+ s,+  . .  .+s„)— =— + +— ^--  •  •  + 

2!  2'-     2  2^  2' 

(n— l)(n— 2) 

aj+(n— l)a,H a.,+  .  .  .+a„ 

2! 


*  L.  D.  Ames.  Annals  of  Mathematics,  Series  2,  Vol.  3,  p.  188.     Equation  (1)  of  thi 
article  should  read  s=:Uj+u,+U3+. . . 

"11 


The  right  member,  which  is  Euler's  form,  is  thus  expressed  in  terms  of  s^,  s^,  S3  . . 
A  Mean  Having  Frohenius'  and  Holder's  Means  as  Special  Cases 

If  we  add to 

m 

S1+S2+  .  .  .  +Sm-i  Si+S.+  .  .  .  +S„ 

,Si,  {  = ,  we  have 


m 


m 


which  is  Frohenius'  mean.f     (When  mS^  or  Frohenius'  mean  has  a  limit,  the 


other  one  has  the  same  limit,  for  mSj: 


Si4-S2+  • . .  +Sm-i  m — 1 


m— 1 


m 


In  general 


"(a,+  ...+a«) 
if  we  add to  mS»  and  form  a  set  similar  to  (B)  upon  the  basis 


m" 


of  this  sum  we  have  a  generalization  of  both  Frohenius'  and  Holder's  t  means, 

n(n±l) 


iOn 


n  s, 


2! 


S2   I    •  •  •  ~rS)i(m_i) 


Add 


n(n±l) 
ai+n  a,H aj-j- . . .  +a, 


2! 


•(«n_i)+i  I 


1/m" 
1/m" 


and  denote  the  sum  by  m(S)i». 
Then 


(26)     „(S)„= 


n(n±l) 

Sj-f-n  SjH ^3~r  '  '  •  "T"Sn(»i-i)+i 


2! 


1/m" 


Analogous  to  (B)  we  formulate; 
(C) 


m(S)„='«[s,+  . .  .+s«]  l/m»=«(S), 
c^3(S)„=»['lHS)«+...+',V(S)„]l/m* 


M(S)„=»[-''l"(S)«+-..4-^^^KS)„]l/m» 

where  n  is  constant  and  m  increases. 

Theorem  VI.     The  means  (C)  satisfy  the  condition  of  consistency  and  sum 
uniformly  convergent  series  uniformly. 


tBromwich:     Loc.  cit.,  Article  51,  Ex.  2. 
JBromwich:     Loc.  cit.,  Article  123. 

12 


From  (21) 

m(S)„=„,S',,+"(a,-f . .  .4-am)l/m'»=m(S')n,  say, 
Then  we  can  write 

S m \^ ) «^^^ S til  ( o  j n 


or 

whence 
or 


"(S+...+S) 


I  (,►5^11  —  S       Hi^o  )n 


m" 


(27) 


"[(s— s,)  +  ...  +  (s— s„,)ll/m"=s— „,(S')„ 
"tRi4-.  •  •+R».]l/m«=s— «(S')„ 


Now,  upon  the  basis  of  (27)  results  for  «,(S')„  and  m(S)„  analogous  to 
Theorem  I,  the  first  part  of  Theorem  II  and  Theorem  III  are  proven  just  as  these 
were.     Theorems  for  (C)  analogous  to  Theorems  IV  and  V  then  follow. 

When  n=l,  (C)  becomes, 

',!,HS)x=[s,+s,+  ...+s,„]l/m 

l^HS),=  [^l'(S),+'^>(S):+...+'''(S)Jl/m 

',:;'(S),=  ['-'(S),+  '-'(S),+  ...  +  '«"(S)Jl/m 

which  is  exactly  Holder's  mean.     Holder,  however,  used   T'i'    where  we  are  using 

'i'(S)^. 

When  n=r^l,  in  (C),  we  have  Frohenius'  mean. 

There  is  an  interesting  relation  between  2(S)n  and  Borel's  original  definition,* 
consequently  between  the  latter  and  Euler's  Transformation.  Horel's  original 
definition  is 

n  x» 

(28)         L        L     ve-s,,^^  — 

X— >oo  n^oo  o  n  ! 


(29)     .,(S)„ 
Introduce  the  factors  1, 
(29)  and  replace  1/2"  by 


n(n-l) 
[Sj+n  s,H S3+ . . .  +s„, J 


X     x^ 

n     n^ 
1 


2.! 

X' 

n» 


1/2" 


as  multipliers  of  the  successive  terms  of 


(1+-)" 
n 


We  then  have,t 


♦Bromwich:     Loc.  cit..  Art.  114. 

t  This   suggests  a   generalization   of   Borel's   original   definition    upon    the   basis   of 
(C),  but  the  resulting  forms  are  exceedingly  complex. 

13 


n(n-l) 
(30)      I  Ls,+n/n  s^x-\ S3  xV2 !+ . . .  +s„^i  x'Vn"] 


(1+-))" 
Denote  this  form  by  s  (x,n). 

Then,  < 

L        L     s(x,n)=     L    [s,-fs,+S3xV2!+...]e-'^ 
x-^00  n— »oo  x— »oo 

n 

=     L        L    2  e-'^Sn+i  xVn ! 
x^oo  n— >oo  o 


while 


L         s(x,n)=r:     L    2(S)n 
x=n— >oo  n— »oo 


Thus  the  sum  by  Borel's  form  and  that  by  2(S)n,  when  they  exist,  can  be 
derived  from  the  same  function,  s(x,n),  the  difference  being  in  the  manner  of 
taking  the  hmits. 

Comparing  (29)  with  (25)  we  see  that 

a^      a^+a,     ai+2a2+a3  "-^(ai+ao)     "(a^+a,) 

,(S)„=-+ + +. . .+ + 

2         2-  2^  2"  2" 

"(ai+aj 

That  is,  o(S),, is  Euler's  Transformation.     Whenever  the  latter 

2« 

<"-^'(a,+a2)  "(a.+a^) 

converges  to  a  limit, converges  to  zero,  hence does.     Con- 

2"  2"^^ 

"(a.+aj 
sequentlv  converges  to   zero.     Hence   when   Euler's   Transformation 

2" 

gives  a  sum,  it  is  the  same  as         L        s(x,n). 

x=n— >oo 

The  means,  (A),  sum  series  for  which  (B)  fails  just  as  Euler's  method 
sums  certain  series  for  which  Holder's  and  Cesaro's  methods  fail.  However,  it  is 
desirable  to  have  different  methods  to  use  on  different  types  of  series  provided  they 
do  not  give  contradictory  results.* 

Illustrations  of  the  Use  of   (A)   and    (B)   on 
Some  Familiar  Series 

*  See  (b)  p.  4. 

14 


(1)  The  series  l4-x+x--|-x^  . . .  which  is  summable  by  (B)  or  Cesaro's  or 
Holder's  methods  only  for  1>X5 — 1  is  summable  by  (A)  for  x<l.  The  actual 
work  follows : 


1— x  1— x- 

Si=lr= ,  S2=1H-X= , 

1— X  1— X 


S„: 


1 X" 


1— X 


xtl 


Use  (A)  with  m=2. 


whence 

1 

L    28','^  = ,  l>x>    (1 — 2'"+^),    where  r  is  arbitrary.     That  is,  the  sum  o 

n— »oo  1 — X 


l-!-x4-x-+. .  .  is for  every  value  of  x<l. 

1— X 

Formulas  (A)  and  (B)  lend  themselves  readily  to  the  actual  calculation  0I 

15 


sums  because  of  the  many  known  relations  between  the  coefficients  in  the  expan- 
sion of  a  power  of  a  polynomial. 

(2)     For  instance  all  series  which  have 

S,=a^  S,=-(2a)^  S,  =  i3ay,  S,=-(4a)^  . . . 
have  the  sum  zero. 


Using  mS^,V  with  m=2, 


C[ll 

2Jn 


n(n-l) 

na* (2a)'^+...  +  (— l)»^Hna)'' 

2! 


=0.n>K* 
For  this  binomial  form  results  from  taking  the  nth  mean  of 

0,  a^— (2a)^  . . .,  (— l)"^^(n  a)^  (see  pp.  6-7). 

ar=l,  K=l,  gives  the  series 

(2')  1— 3-f5— 7-f...=0 

a=:l,  K=:2  gives  the  series 

(2")  1— 5-fl3— 25-j-...=0 

(3)     The  series 
1—1-1-1— 1-f 

sums  to  1/2  by  (A)  using  any  m  or  by  (B)  using  any  n,  or  by  either  with  m  and  n 
both  increasing. 
Here 

Si=:l,  So^O,  83=1,  ... 


.S'!'^ 


n+0+...-f 


(l-\-( l)»(»i-i)+i 


l/m"=„,Sn 


When  m  is  even  the  sum  of  the  even  coefficients  in  (l-|-x-|-. .  .-j-x'""^)" 
equals  the  sum  of  the  odd,  and  when  m  is  odd,  the  sum  of  the  even  equals  the 
sum  of  the  odd  minus  unity.  In  either  case  if  O  and  E  denote  the  sum  of  the 
odd  and  of  the  even  coefficients,  respectively, 


E+0=r2E- 


l_|_(_l)m.l 


That  is, 


m«=2E- 


l_|-(_l)m.l 


*  A.  M.  Kenyon:     "Some  Properties  of  Binomial  Coefficients,"  Proceedings  of  the 
Indiana  Academy  of  Science,  1914,  p.  2. 

16 


Whence 


m" 


E: 


l4-(— l)«-i 


2  4 

Substituting  this  value  of  E  in  the  above  form  of  S, 


1 
2 


l  +  (— l)*"^^ 


m" 


1/m" 


Hence 


(4)     The  series 


n— >oo  m— >oo  »»,n— »oo  2 

a— (a+d)  +  (a+2d)— (a+3d)  +  . . . 
2a— d 


stuns  to 


,On — 


s,=:a,         s.r=d,         s^=a+d.         s^^ — 2d,         S5=a4-2d,  . . . 

n(n— 1)        n(n— l)(n— 2)                    n(n— 1)  (n— 2)  (n— 3) 
n  a ■ — d-\ (a+d) ^2d 


2! 


3! 

n...(n— 4) 


4! 


■(a+2d)-... 


1 

2» 


1  rn(n— 1)        n(n— l)(n— 2)     n(n— 1)  (n— 2)  (n— 3) 

_9" a) I — — : 

3! 
n. . .  (n — 4. 


a— 2"— d-^ 1 -2 

2  12!  3!  4! 


-2.. 


5! 


1 
2- 


SimpHfy  the  coefficient  of  a.     In  the  coefficient  of  d  add  and  subtract 

Then  collect  one-half  of  the  sum  of  the  even  binomial  coefficients. 
This  gives, 


2       d 

2       2» 


n     ln(n— l)(n— 2) 

-+ 

2    2  3! 


11 

d 

+- 

2 

2" 

— 1       2  n(n— 1) 

n+ 

2       2       2! 


3n(n— l)(n— 2)     4n...(n— 3)      5n...(n— 4)  n 

+ +...  +  (_!)«_ 

2  3!  2         4!  2        5!  2 

17 


a       d  2» 


2       2»  2'       2 


n+l 


n(n— 1)         n(n— l)(n— 2) 
o — n+2 3- 


2!  3! 


n...(n— 3) 

4!  J 

The  bracket  is  zero,  n  5  2,  since  it  results  from  taking  the  nth  mean  two  at  a 

time  of  the  terms  o,  1,  — 2,  3,  — 4,  5,  . . .,  ( — l)"n,  (see  pp.  4-6). 

Whence, 

2a— d 

2^H— —  — —        ■L<        2^'> 

4        n— >oo 
(4')     ar=l,  d=l  gives 

1—2+3— 4+ 5...  =  14 
(4" )     a=  1 ,  i\-^2)  gives 

l-4+7-10+...=->4 
It  is  interesting  to  notice  that  the  sum  of 

a— (a+d)  +  (a+2d)— (a+3d)  +  .  .  . 

< 

is  positive,  zero  or  negative  according  as  d  =  2a.  (See  (2'),  p.  16.) 

> 

This  result  could  have  been  obtained  by  using  (B). 
3.     Non-summable  Series.* 
Consider  the  series 

(1)  a,+a.,-fa3+...+a„+... 

and  the  sequence 

n 

(2)  Sj,  s...  S3,  . . . ,  s„,  ...  where  s„=r:2a„ 

1 

Definition.     Denote  by  N.,  the  class  of  series  which  are  such  that  for  every  c 
and  every  M  there  exists  an  m  5  M  such  that  for  every  n  5  1, 

n 

2    (s;„,p— c)5  0,  (orgO). 

P=l 

The  class  N..  contains  all  properly  divergent  series  and  such  series  f  as 


*  Presented  to  the  American  Mathematical  Society,  April  29,  1916. 
fThe  sequences  of  S's  are  respectively:    1,  0.  2,  0,  3,  0,  4.  0,  ...  and  2,  —7.  4.  — 2, 
6,  — 3,   ....     In  either  case  for  a  given  C  we  can  choose  m  such  that  S,„^j+S„,^^>C, 

n 
consequently     2     (S„,^„ — C)50,  n^l. 
p=l 

18 


1-1+2-2+3-3+... +  (-1)' 


2n+l+(— l)«*i 


2—3+5—6+8—9+ . . .  +  (—1 )  "^'- 


6n+l  +  (— l)"-i 


The  method  of  procedure  in  this  article  is  to  place  sufficient  restrictions  upon 
the  general  definition,  (1),  §  1,  to  make  the  members  of  N«  nonsummable.J,  then 
prove  upon  the  basis  of  this  general  method,  that  the  specific  methods  mentioned 
in  §  1  can  not  sum  series  in  the  class  N.,.  It  is.  incidentally,  shown  that  the  general 
definition  thus  restricted  satisfies  the  condition  of  consistency  and  sums  uniformly 
convergent  series  uniformly. 

Definition  (1)  §  1  can  be  expressed  in  terms  of  s^,  s^,  ...   s„.   ...  as  follows: 

(3)     S„=fia,  +  f2a.+  . .  .+f»_ia„_,+f„a„. 


(3') 
(3") 
(4) 
where 

whence 
(5) 


-fiSr+l.(S,— Sj)  +  .  .  .+f„_i(s„_i— S„_,)+f„(S„— S„..j) 

^Si(f  1— f2)+s,(f  — f,)  +  ...+s„.,(f„.,— f„)+sj„ 

:Sj*i+S2<I'2+-  •  .+S„.i*«_i  +  S„f„ 


it'—h.i=^l>, 


p=l,2,  3,  ...,n— 1, 


S=      L        L 


n— 1 

2     Sp«^p(Xi)+S„fn 
1 


i=l,2,  3,  ...  k 


Now  let 

(6)     0p=s,a — c,  (see  definition,  p.  18),  and  replace  s,,  by  c+0p  in 

n— 1 

S„=    2    Sp<l>p+s„f„ 
1 


Then 


(7) 


n —  1         n — 1 

S„=c    2   <^„+    2    0p^,+s,.f„ 
1  1 


But 


n— 1         n— 1 

2    $,=    2    (f„_,-f,)  =  f,-f,. 
1  1 


t  These  restrictions  are  chosen  so  as  to  include  in  the  restricted  general  method 
those  special  methods  by  which  the  class  of  series  N^  are  non-summable,  these  specific 
results  having  been  obtained  independently. 

19 


Whence 


n— 1 

S„=cfi+    2    0p*p+(s„— c)f„ 
1 


or 


m  n — 1 

(8)  S„=cf,+2®p<I»p+    2   0,$p+®«fn 

1  m+1 

where  m  is  arbitrary  and  n  5  m+2. 

Consider,  first,  the  terms 

m 
c  f  1+  2  ®p% 
1 

If  we  assume 

(9)  L      fp(x,)=:l,  (i=l,  2,  ...,  k),  for  every  p, 
Xi-^Li 

then  L     *p=      L      (f„.i — fp)=o 

whence 

m 

L      (cfi+2  0p*p)=c 
Xj-»L(  1 

n— 1 
Now  consider  the  term  2    0p*p 

m+1 

n— 1  n— 1 

Let    s'„=    2    0p.     Consider  c  positive  and      2    ©p  >  o    for   n>m+l>M. 
m+1  m+1 

(See  definition,  p.  18.) 

Then  in  the  Hght  of  (4)  we  can  write 

n— 1 

(10)  2     ®„^p=s'm^i(^m^i *»i+2)+s'».+2(*m+2 ^»U3)  +  --'- 

m+1 

+  S'„_2  ($„_2 *„_i  )  +S'„_i*„.i 

n— 1 

and    2    ®p%  will  be  positive  or  zero  provided  (^m+p — ^m+p^i), 
m+1 

(p=:l,  2. . . ,  (n — m — 2)  and  ©„.i,  are  positive  or  zero.    This  will  be  the  case  if  we 
assume  that 

20 


(U)     ^m^p,  [=f„,^p(Xi) — f,».p+i(xj)],  P^O,  never  increases  with  p,  (for  any 
set  of  values  of  Xi,  i=l,  2,  3,  ...  k),  M  having  been  properly  chosen, 
and  that 

(12)  L     f„(xi)=o,  for  all  values  of  x^. 
n— »oo 

For  from  (12)       L    4>„=o  and  from  this  fact  and  (11), 
n^oo 

(13)  *„_,=  (f„.,— f„)5  0,  n5M+l 
Finally  consider  the  last  term  of  (8),  namely  0„f„.     Evidently 

(14)  f„(xi)5  0,  n^M+l. 

Hence  0„f„  can  not  remain  negative,  for  0„  can  not  remain  negative  and 

n 
increase  without  limit  in  absolute  value  since     2     @m+p  5  o,  n  5  1. 

P=l 

We  have  thus  shown  that  S,,  either  oscillates  or  can  be  made  greater  than  any 
positive  quantity  C,  by  a  proper  choice  of  m. 

n 
The  argument  for  series  such  that     S     (s„,^p — c)  ^  o,  is  the  same  as  the  above. 

P=l 

Theorem  VII.     Every  series  of  the  class  N«  is  non-summable  by  the  defini- 
tion 

n 

S=      L  L       2    a„f,(xO,  i=1.2.  . . .  k, 

Xi— >Lt  n— >oo  p=l 

where 

I.  L      ip{xi)  =  \,  for  every  p 

Xi-^Li 

II.  L     f„(xi)=o 

n— >oo 

and 

III.     For  every  set  of  values  of  x^,  x^,  . . .,  xt,  there  exists  an  M  such  that 
the  sequence, 

(  Im+p I»i+p+ij)  P^— ^'J;  A?  ^>   •  •  • 

never  increases. 

Theorem  VIII.     Definition  (1)  §  1  under  restrictions  I,  II  and  III  satisfies 
the  condition  of  consistency. 

Let  a,-|-a2+ . . .  +an4-  •  •  •  converge. 

21 


Since,  s — Ri=Si,  s — Ra^Sg,  ...,  s — R„=:s„ 

where 

the  form  of  s„  in  (3")  (p.  19),  can  be  written,  after  collecting  the  coefficients  of  s, 

n— 1 

Su^sfi 2     Rp(fp— fp^.l) — Rnf» 


and  for  any  m,  (^n — 1),  we  can  write 


S„=s  f,— 2R,(fp— f,,J- 
1 


By  I  of  Theorem  VII, 

L 

x«— >Li 


n— 1 

2      Rp(f,— f,,J+R«fn 

m+1 


sf— 2Rp(fp— f,,,) 
1 


=  s 


The  last  term  on  the  right  converges  to  zero  for 

"n— 1 

5    Rp(fp— fp,,)+R,.f„ 
m+1 


<€ 


Im+i       Im+2"T~  tni+2        •  •  •~il>i 


provided  m  is  chosen  large  enough  to  make  III,  Theorem  VII,  hold  and  also  to 
make  |Rp|<c,  p>m. 

Now  from  II  and  III,  there  exists  an  m  such  that  * 
o^fp<l,  p>m 

whence  there  exists  an  m  such  that 


n— 1 

2      Rp(fp fp+i)+Rnf»i 

m-f-1 


<€, 


n>m+l. 


Corollary.     The  general  definition  restricted  by  I,  II  and  III  sums  uniformly 
convergent  series  uniformly. 

Theorem  IX.     Le  Roy's  method  can  not  sum  series  in  the  class  Ng. 

Le  Roy's  f  definition  is 


(15) 


n    r(pt+l) 

:    L      L       2   ap, 

t— >1  n— >co  p=o  r(p-|-l) 


l>t>o. 


♦Choose   an  m  such  that  |fm+pl<Cl>  P>o-     Then  the  sequence,  f„,,  f,„^^ never  in- 
creases and  converges  to  zero  according  to  II  and  III.     Hence  f„,^p>o. 
t  Annates    de  Toulouse,  Ser.  2,  Vol.  2  (1900),  pp.  323-327. 

22 


Comparing  with  the  general  definition  of  Theorem  VII  we  see  that 

r[(p-l)t+l] 

fp(Xi,  X2,    .  .  .,  Xfc)  =  fp(t)= 

r(p) 

Condition  I  of  the  above  theorem  is  satisfied  since  . 

r[(p-l)t+i] 

L   =1 

t-.l        r(p)) 

r[(n-i)t+i] 

Condition  II  holds  for      L      =0*,  l>t>0. 

n^oo  r(n) 

In  order  to  show  that  Condition  III  holds  it  must  be  shown  that  for  every 
t,  l>t>0,  there  exists  a  P  such  that  for  every  p>P, 

r(pt+l)     r(pt+l+t)  r(pt+l+t)       r(pt+l+2t) 

(16) % 

•r(p+i)        r(p+2)        r(p+2)  r(p+3) 

Since  p  is  a  positive  integer,  the  denominators  are,  respectively,  p!,  (p+1)  ! 
and  (p+2)  !     By  the  identity  r(x+l)=xr(x), 

r(pt+l+t)   r=(p+l)tr(pt+t), 
and 

r(pt+l+2t)=(p+2)tr(pt+2t) 

whence  (16)  reduces  to 

t 

r(pt+l)— tr(pt+t)  5  tr(pt+t) r(pt+2t) 

P+l 


or 


r(pt+l) 

t5t 

r(pt+t) 


1  — t- 


r(pt+2t) 
1 

(p+l)r(pt+t)J 
r(pt+2t) 


r(pt+l+t) 

Since  the  right  hand  side  is  less  than  t  it  will  be  sufficient  to  show  that 

r(pt+l) 


r(pt+t) 


5  2t 


♦Smail:     Dissertation,  Columbia  University,  1913,  pp.  38-39. 

23 


By  Sterlings  formula,* 

r(pt+l)         e-^pf^i-''Upt-\-t—l)p*^t-^/^V2ll 

(17)         L — =1 

p-^oo  e-p'(pt)p'^i/-V2n  r(pt+t) 

Or  simplifying, 

r(pt+l)  t— 1 

(18)         L    e^-'(lH )^'-^/-(pt+t— l)'-^=rl 

p-^oo  r(pt+t)  pt 


But 


and 


t— 1 

L     (1-^ yui/2^^t~i 

p-^oo  pt 


L     (pt+t — l)'-^=:o,  since  t<l. 
p— »oo 

Thus  the  limit  of  the  coefficient  of  r(pt4-l) 

in  (18)  is  zero. 

r(pt+t) 
Hence, 

r(pt+i) 

must  increase  without  limit  as  p  increases. 

r(pt+t) 

which  proves  (16). 

Thus  LeRoy's  method  has  been  shown  to  be  a  special  case  of  the  general 

definition  restricted  as  in  Theorem  VII.     Hence  both  Theorem  VII  and  Theorem 
VIII  t  hold  for  it. 

Theorem  X.     Neither  Borel's  integral  definition  5=/^  e--''u(x)dx  nor  his 

original  definition  S=     L      2  S,, can  sum  series  of  the  class  N,. 

X— >  00   "  p ! 

If  Borel's  original  definition  sums  a  series,  his  integral  definition  gives  the 
same  sum,J  and  if  Borel's  integral  definition  sums  a  series,  LeRoy's  method  gives 
the  same  sum.  §  By  Theorem  IX.  LeRoy's  method  can  not  sum  series  of  the  class 
N,.  hence  neither  of  Borel's  methods  can.^ 


*Bromwich:     Loc.  cit.,  page  462. 
t  See  note,  page  28. 
JBromwich:       Loc.  cit..  Article  114. 
§  Bromwich  :      Loc.  cit.,  Article  116. 

^  It   can    be    shown    by  an   argument   similar   to   that   of   Theorem    IX    that    Borel's 
methods  are  special  instances  of  the  general  method,  see  note.  p.  26. 

24 


Theorem  XI.     Cesaro's  mean  *  5=      L    


n-^oo  A^";' 

r(r+l)  r(r+l)...(r+n— 1) 

where  S^r,^  =s„+r  s„_.^-\ S„.2+  •  •  •  H So 

2!  n! 

and  A':;'  =  (r4-l)(r+2)...(r4-n)/n!, 
can  not  sum  series  of  the  type  Nj. 

First  consider  r  5  1. 
Sn  can  be  written 

n  n(n — 1)  n  la^ 

( 19)     a„+ a,+ a,+ . . .  -f 


r+n         (r+n— l)(r+n)    '  (r+1)  (r+2) .  . .  (r+n) 

tivei 
)n  c 
Then.t 


It  is  convenient,  now,  to  let  p  run  from  o  to  n,  instead  of  1  to  n,  in  the  gen- 
eral definition  of  Theorem  VII.     This  complies  with  the  usage  in  Cesaro's  mean. 


n(n— l)...(n— p+1) 

(20)      fp(xO  =  f,(n)= .  P^n 

(r+n— p+1 )  (r+n— p+2) . . .  (r+n) 


Now 

fp(n)  may  be  written, 


n — i  n — p+1 


r+n — p+1        (r+n — p+2)  r+n 

Each  of  these  fractions  approaches  unity  as  n  increases,  and  there  are  exactly 
p  of  them.     Hence, 

(21)  L     fp(n)=rl,  for  every  p.$ 

n— >oo 

that  is.  condition  I  of  Theorem  VII  holds. 
Again, 


h(n): 


(r+1)  (r+2).  (r+n)     r+1  r+2       r+n— 1     n+r 


*Bullitin    des  Sciences  Math.  (2),  V.  14  (1890),  p.  119. 
t  In  this  case   x^.^n.     See  note,  p.  1. 
ifln  this  mean  fo(n)r=:l. 

25 


1 

Every  fraction  on  the  right  is  equal  to  or  less  than  unity  and has  the 

n+r 
limit  zero.     Hence 

(22)  L     f„(n)=o, 

n-^oo 

that  is,  condition  II  is  satisfied. 

Condition  III.     From  (19)  it  is  easily  seen  that  ip — (p^^,  p=0,  1.  2,   . . .,  is, 

* 
r  rn  rn(n — 1) 


r+n    (r+n— l)(r+n)     (r+n— 2)  (r+n— 1)  (r+n) 
These  terms  never  increase  when  r  5  1. 

Finally  it  is  necessary  to  note  that  ip — f,,+,  is  always  positive  since  (13),  page 
21,  does  not,  in  this  case,  follow  from  conditions  II  and  III.     It  would,  also, 

suffice  to  prove      L     f„_i(n)  =0. 
n— >oo 

If  Cesaro's  mean  sums  a  series  with  any  value  of  r,  it  gives  the  same  sum  with 
any  greater  r.*     Hence  Cesaro's  mean  can  sum  series  in  N.,  for  no  value  of  r.f 
Theorem  XII.     Holder's  mean  can  not  sum  series  of  the  class  Ns. 
It  has  been  shown  by  Knopp  $  that  Holder's  mean 

1  sv; 

V'J= — -So+s,+  . .  .+s„)=- 


n+l  A';; 

1 

Vf}= (T*''  +T7  + . . . +T T) 

n+1 

1 

'r;>= — (x<'-j'>+T"-:"+. .  .+T"'-'0 

n+l 

gives  a  sum  only  when  Cesaro's  gives  the  same  sum.     Thus  the  above  theorem  is 
a  consequence  of  Theorem  XI. 

The  means  §  (A),  (B)  and  (C)  of  article  2  do  not  satisfy  condition  III  of 
Theorem  VII  for  the  polynomial  coefficients  first  increase  and  then  decrease  and 
the  maximum  coefficient  moves  out  the  sequence  s^,  Sj,  ...  as  n  increases.  Hence 
we  cannot  choose  an  m  such  that  fm+p — fm+p+i  never  increases  as  p  increases. 


*  Chapman,  Proc.'s,  London  Math'l  Soc,  Vol  9  (1911),  pp.  369-409. 

fWe  have  as  a  corollary  of  Theorems  IX,  X  and  XI,  that  LeRoy's,  Borel's  (see 
note,  p.  24)  and  Cesaro's  methods  satisfy  the  condition  of  consistency  and  sum  uni- 
formly convergent  series  uniformly  but  these  are  familiar  results. 

:|:K.   Knopp,   Inaugural  Dissertation  (Berlin,  1907),  p.  19. 

§  The  question  of  the  non-summability  of  series  in  N,  by  these  means  will  be  con- 
sidered in  a  later  paper. 

26 


BIBLIOGRAPHY 

Ford,  "Studies  on  Divergent  Series  and  Summability,"  Michigan  Science 
Series,  Vol.  II.  This  treatise  contains  an  extensive  bibhography  to  which  I  will 
only  add, 

L.  L.  Smail's  Dissertation,  "Some  Generalizations  in  the  Theory  of  Divergent 
Series,"  Columbia  University,  1913. 


27 


VITA 

Glenn  James  was  born  October  2nd,  1882 ;  attended  Vincennes  University 
1901-1903;  received  the  A.  B.  degree  from  Indiana  University  1905  and  the 
A.  M.  1910;  studied  at  Chicago  University  two  summers,  and  at  Columbia  one. 
summer  and  throughout  the  year  1915-1916;  instructor  at  the  Michigan  Agricul- 
tural College  1905-1908  and  at  Purdue  University  since  then;  member  of  The 
American  Mathematical  Society  and  the  Indiana  Academy  of  Science ;  has  pub- 
lished "The  Accuracy  of  Interpolation  in  a  Five  Place  Table  of  Logarithms  of 
Sines"  in  The  American  Mathematical  Monthly,  October.  1913,  in  conjunction 
with  Professor  A.  M.  Kenyon,  and  "A  Note  on  the  Sum  of  the  Remainders  of  a 
Convergent  Series,"  American  Mathematical  Monthly,  November,  1916.  He 
wishes  to  express  his  gratitude  to  Professor  W.  B.  Fite  for  the  inspiration  received 
from  him  and  his  very  careful  criticism  of  this  dissertation. 


28 


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